the Molecular Velocities in Gases. 51 



A 31'. The latter can moreover take all values from to 



2> while the other Taries from to ir. 



Let ©' be a surface-element situated in the upper hemi- 

 sphere, and comprised on the one side between two meridians 

 very near one another and making the angle d<f>, on the other 



■ between two horizontal circles having * and i+di for their 

 angular distances from the point A, and consequently p sine- 

 tor radius; we can, as is known, assimilate co' to a small rect- 

 angle paving for its sides pdi, p sin idcp, and for its surface 

 SiTiT M . ' ^ The P oints M corresponding, as above, to 

 all the points W situated in the area co' evidently constitute 

 an element co, comprised between the same meridians and be- 

 tween two horizontal circles having, for their angular distances 

 from the pomt A, 2i and 2(i + di), and therefore for their sur- 

 faces 



co = 2p- sin 2i . di dcp, or co = ±co' cos i. 



By drawing a vertical of the length Yt through each point 

 of the area a/ we i shall form a very thin prism comprised in 

 the cvlmder of which we have spoken, in the interior of which 

 the centre G of a molecule must be in order that the typical 

 point of the velocity due to the collision mav be within the 

 element co. Its horizontal section, the plane "of which makes 



■ the angle i with the tangent plane in W, will consequently be 



cost or j- 



•If now co designates not an element, but a finite portion of 

 tne sphere, it can be decomposed into elements such a* the 

 preceding; and for the velocity due to the collision to corre- 

 spond to this new region, the point G must be interior to one 

 or the other of the small prisms corresponding to each element, 

 or, what comes to the same thing, within the second cylinder 

 formed by their union, the section of which is again -. Sup- 

 posing the construction of this second cylinder identically 

 repeated for all the molecules of the second kind, the number 

 M> ox the collisions for which the new velocity correspond* to 

 the region co will be the number of the centres G of descend- 

 ing molecules situated in the interior of the second cylinder 

 and oi its homologues. 



Kow p, was the number of points G interior to the first cylin- 

 der and its homologues; £ is therefore the ratio of their 



B2 



